Find all subsets of $\mathbb N$ like $S$ such that \[\forall m,n \in S \implies \dfrac{m+n}{\gcd(m,n)} \in S \]

Peter and Paul gamble as follows. For each natural number, successively, they determine its largest odd divisor and compute its remainder when divided by $4$. If this remainder is $1$, then Peter gives Paul a coin; otherwise, Paul gives Peter a coin. After some time they stop playing and balance the accounts. Prove that Paul wins.

Prove that there exist two natural numbers $a,b$ such that $|a-m|+|b-n|>1000$ for any relatively prime natural numbers $m,n$.

a) A positive integer is written at each vertex of a triangle. Then on each side of the triangle the greatest common divisor of its ends is written. It is possible that the numbers written on the sides be three consecutive integers, in some order? b) A positive integer is written at each vertex of a tetrahedron. Then, on each edge of the tetrahedron is written the greatest common divisor of its ends . It is possible that the numbers written in the edges are six consecutive integers, in some order?

Prove that if $ n $ is an integer greater than $ 4 $, then $ 2^n $ is greater than $ n^2 $.

A quadratic trinomial $p(x)$ with real coefficients is given. Prove that there is a positive integer $n$ such that the equation $p(x) = \frac{1}{n}$ has no rational roots.

Given are positive integers $n>20$ and $k>1$, such that $k^2$ divides $n$. Prove that there exist positive integers $a, b, c$, such that $n=ab+bc+ca$.

If $ab(a+b)$ divides $a^2 + ab+ b^2$ for different integers $a$ and $b$, prove that \[|a-b|>\sqrt[3]{ab}.\]

Let $a$ be a positive integer, but not a perfect square; $r$ is a real root of the equation $x^3-2ax+1=0$. Prove that $ r+\sqrt{a}$ is an irrational number.

Let $a$ and $b$ is roots of the $x^2-6x+1$ equation. [b]a[/b]) Show that, for all $n \in{\mathbb Z^+}$ , $a^n+b^n$ is a integer. [b]b[/b]) Show that, for all $n \in{\mathbb Z^+}$ , $5$ isn't divide $a^n+b^n$

$15.$ Let $n$ be the largest integer that is the product of exactly $3$ distinct prime numbers, $x,y,$ and $10x+y,$ where $x$ and $y$ are digits. What is the sum of digits of $n ?$

Let $n,m,r$ be positive integers such that $n>m$ and both $n^2+r, m^2+r$ are powers of $2$. Show that $n>\frac{2m^2}{r}$.

If $n$ is a positive integer such that $8n+1$ is a perfect square, then $\textbf{(A)}~n\text{ must be odd}$ $\textbf{(B)}~n\text{ cannot be a perfect square}$ $\textbf{(C)}~n\text{ cannot be a perfect square}$ $\textbf{(D)}~\text{None of the above}$

Let $P$ be the set of all primes. Given any positive integer $n$, define $$\displaystyle f(n) = \max_{p \in P}v_p(n)$$ Prove that for any positive integer $k\ge 2$, there exists infinitely many positive integers $m$ such that \[ f(m+1) = f(m+2) = \cdots = f(m+k) \] [i]Proposed by Ivan Chan Guan Yu[/i]

Prove that: [b](a)[/b] If $(a_n)_{n\geq 1}$ is a strictly increasing sequence of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}$ is a constant as $n$ runs through all positive integers, then this constant is an integer greater than or equal to $4$; and [b](b)[/b] Given an integer $N\geq 4$, there exists a strictly increasing sequene $(a_n)_{n\geq 1}$ of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}=N$ for all indices $n$.

Find the largest possible value of the positive integer $N$ given that there exist positive integers $a_1, a_2, \dots, a_N$ satisfying $$ a_n = \sqrt{(a_{n-1})^2 + 2018 \, a_{n-2}}\:, \quad \text{for } n = 3,4,\dots,N. $$

[u]Round 1[/u] [b]p1.[/b] Compute $(1 - 2(3 - 4(5 - 6)))(7 - (8 - 9))$. [b]p2.[/b] How many numbers are in the set $\{20, 21, 22, ..., 88, 89\}$? [b]p3.[/b] Three times the complement of the supplement of an angle is equal to $60$ degrees less than the angle itself. Find the measure of the angle in degrees. [u]Round 2[/u] [b]p4.[/b] A positive number is decreased by $10\%$, then decreased by $20\%$, and finally increased by $30\%$. By what percent has this number changed from the original? Give a positive answer for a percent increase and a negative answer for a percent decrease. [b]p5.[/b] What is the area of the triangle with vertices at $(2, 3)$, $(8, 11)$, and $(13, 3)$? [b]p6.[/b] There are three bins, each containing red, green, and/or blue pens. The first bin has $0$ red, $0$ green, and $3$ blue pens, the second bin has $0$ red, $2$ green, and $4$ blue pens, and the final bin has $1$ red, $5$ green, and $6$ blue pens. What is the probability that if one pen is drawn from each bin at random, one of each color pen will be drawn? [u]Round 3[/u] [b]p7.[/b] If a and b are positive integers and $a^2 - b^2 = 23$, what is the value of $a$? [b]p8.[/b] Find the prime factorization of the greatest common divisor of $2^3\cdot 3^2\cdot 5^5\cdot 7^4$ and $2^4\cdot 3^1\cdot 5^2\cdot 7^6$. [b]p9.[/b] Given that $$a + 2b + 3c = 5$$ $$2a + 3b + c = -2$$ $$3a + b + 2c = 3,$$ find $3a + 3b + 3c$. [u]Round 4[/u] [b]p10.[/b] How many positive integer divisors does $11^{20}$ have? [b]p11.[/b] Let $\alpha$ be the answer to problem $10$. Find the real value of $x$ such that $2^{x-5} = 64^{x/\alpha}$. [b]p12.[/b] Let $\beta$ be the answer to problem $11$. Triangle $LMT$ has a right angle at $M$, $LM = \beta$, and $LT = 4\beta - 3$. If $Z$ is the midpoint of $LT$, what is the length$ MZ$? PS. You should use hide for answers. Rounds 5-8 are [url=https://artofproblemsolving.com/community/c3h3133709p28395558]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134133p28400917]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer. There is a collection of cards that meets the following properties: $\bullet$Each card has a number written in the form $m!$, where $m$ is a positive integer. $\bullet$For every positive integer $t\le n!$, it is possible to choose one or more cards from the collection in such a way $\text{ }$that the sum of the numbers of those cards is $t$. Determine, based on $n$, the smallest number of cards that this collection can have.

Let $a,p,q\in \mathbb{Z}_{\ge 1}$ be such that $a$ is a perfect square, $a=pq$ and $$2021~|~p^3+q^3+p^2q+pq^2.$$ Prove that $2021$ divides $\sqrt a$.\\ \\ [i](Cosmin Gavrilă)[/i]

Determine all real numbers $a, b$ and all integers $n\ge 1$ for which$ (a + b)^n = a^n + b^n$ holds.

Let $ A$, $ M$, and $ C$ be digits with \[ (100A \plus{} 10M \plus{} C )(A \plus{} M \plus{} C ) \equal{} 2005. \]What is $ A$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

For which $n{}$ is it possible that a product of $n{}$ consecutive positive integers is equal to a sum of $n{}$ consecutive (not necessarily the same) positive integers? [i]Boris Frenkin[/i]

Let $p>3$ be a prime number and $n=\frac{2^{2p}-1}3$. Show that $n$ divides $2^n-2$.

Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$. Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$. (a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $(a, b)$ with $a\mid b$. (b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.

A natural number $n$ is called "good" if there exists natural numbers $a$ and $b$ such that $a+b=n$ and $ab \mid n^2+n+1$. Show that there are infinitely many "good" numbers